An integrable case of a rotational motion analogous to that of Lagrange and Poisson for a gyrostat in a Newtonian force field

Abstract

The scope of the present paper is to provide analytic solutions to the problem of the attitude evolution of a symmetric gyrostat about a fixed point in a central Newtonian force field when the potential function isV ⁽²⁾.

We assume that the center of mass and the gyrostatic moment are on the axis of symmetry and that the initial conditions are the following: ψ(t ₀)=ψ₀, θ(t ₀)=θ₀, ψ(t ₀)=φ(t ₀)=φ₀, ω₁(t ₀)=0, ω₂(t ₀)=0 and ω₃(t ₀)=ω ⁰₃ .

The problem is integrated when the third component of the total angular momentum is different from zero (B ₁ ≠ 0). There now appear equilibrium solutions that did not exist in the caseB ₁=0, which can be determined in function of the value ofl ^r₃ (the third component of the gyrostatic momentum).

The possible types of solutions (elliptic, trigonometric, stationary) depend upon the nature of the roots of the functiong(u). The solutions for Euler angles are given in terms of functions of the timet. If we cancel the third component of the gyrostatic momentum (l ^r₃ =0), the obtained solutions are valid for rigid bodies.